GKM actions on almost quaternionic manifolds

TL;DR

This paper develops a combinatorial framework for GKM graphs with quaternionic structures, characterizing certain GKM actions on positive quaternion-Kähler manifolds and classifying the associated graphs.

Contribution

It introduces quaternionic structures on GKM graphs and classifies GKM$_3$ actions on specific quaternionic manifolds using combinatorial properties.

Findings

GKM$_3$ graphs with quaternionic structures correspond to actions on quaternionic projective space and Grassmannians.

Quaternionic 2-faces are biangles or triangles, while complex 2-faces are triangles or quadrangles.

Abstract GKM$_3$ graphs with these face restrictions are classified as arising from specific quaternionic manifolds.

Abstract

We introduce quaternionic structures on abstract GKM graphs, as the combinatorial counterpart of almost quaternionic structures left invariant by a torus action of GKM type. In the GKM$_3$ setting the 2-faces of the GKM graph can naturally be divided into quaternionic and complex 2-faces; it turns out that for GKM$_3$ actions on positive quaternion-K\"ahler manifolds the quaternionic 2-faces are biangles or triangles, and the complex 2-faces triangles or quadrangles. We show purely combinatorially that any abstract GKM$_3$ graph with quaternionic structure satisfying this restriction on the 2-faces of the GKM graph is that of a torus action on quaternionic projective space ${\mathbb{H}} P^n$ or the Grassmannian ${\mathrm{Gr}}_2({\mathbb{C}}^n)$ of complex 2-planes in ${\mathbb{C}}^n$.